Type transformations for sharp characters(Cohomology Theory of Finite Groups and Related Topics)

35

Type

transformations

for

sharp

characters

Masao KIYOTA

College of Liberal Arts and Sciences Tokyo Medical and Dental University

1

Introduction

Let $G$ be

a

finite group and $\chi$ be

a

faithful character of$G$ ofdegree $n$

.

Put

$L=\{\chi(g)|g\in G, g\neq 1\}$

.

Then

we

have the following

Theorem 1 (Blichfeldt[B])

_|G|

divides the integer $\prod_{l\in L}(n$-l).

Theorem 1 gives

us

the upper bound of the order of $G$. We

are

interested in

the

case

$G$ attains the bound.

Definition 1 We call (G,$\chi)$ sharpoftype L (or L sharp if _{$|G|= \prod_{l\in L}(n-l)$}

holds.

Problem 1 For

a

given L, determine all $L$-sharp pairs (G,$\chi)$.

Example 1 Let $G$ be

a

sharply $t$-transitive permutation group, which is

different from $S_{t}$, the symmetric group of degree $t$

.

Let $\pi$ be the permutation

characterof$G$. Then $(G, \pi)$ is sharp oftype $\{0, 1, \cdots, t-1\}$.

Note that $(G, \chi)$ is sharp ifand only if $(G, \chi+1_{G})$ is sharp, where $1_{G}$ is the

trivial character of$G$.

So we

may

assume

$(\chi, 1_{G})=0$ holds, when

we

consider

sharp characters $\chi$. We call such character normalizedsharp character.

We have the following results concerning Problem 1. When $L$ contains

an

irrational number, $L$-sharp pairs $(G, \chi)$

are

completely classified by

Alvis-Nozawa[A-N]. Hence

we

may

assume

that $L\subset \mathrm{Z}$ holds. The

cases

$L=$ $\{l\}$, $\{l, l+1\}$, $\{l, l+2\}$, $\{l, l+1, l+2\}$, $\{l, l+1, l+2, l+3\}$

are

treated in

Cameron-Kiyota [C-K], Cameron-Kataoka-Kiyota $[\mathrm{C}- \mathrm{K}-\mathrm{K}]_{\mathrm{Y}}$ Nozawa [N], We

do not have any classification results for ”big” $L$ in

case

$L\subset \mathrm{Z}$, and

so

we

should ask the following

Problem 2

Can

we

reduce the

classification

of$L$-sharp pairs to that of $L’-$

sharp pairs for

some

$L’$ with _$|L’|<|L|$ ?

38

2

Transformations

of

types

Let $L_{1}$, $L_{2}$ be finite sets of complex numbers with $|L_{1}|=|L_{2}|=m\geq 2$

.

Definition2 Wewrite$L_{1}\sim L_{2}$if$e_{1}(L_{1})=e_{1}(L_{2})$, $e_{2}(L_{1})=e_{2}(L_{2})$, $\cdots$ ,$e_{m-1}(L_{1})=$

$e_{m-1}(L_{2})$ hold, where $e_{k}(L_{1})$ is the k-th elementary symmetric function with

variables in $L_{1}$. For example, $e_{1}(L_{1})= \sum_{l\in L_{1}}l$, $e_{m}(L_{1})= \prod_{l\in L_{1}}l$.

Example 2

{a,

$b\}\sim$

{c,

$d\}\Leftrightarrow a+b=c+d$,

{a,

b,$c\}\sim$

{d,

e,$f\}\approx$

$a+b+c=d+e+f$

, $ab+bc$$+ca=de+ef+fd$

The following two lemmas

are

fundamentalbut easy to prove. Lemma 1

(1) $L_{1}\sim L_{2}\Leftrightarrow L_{1}+l\sim L_{2}+l$, where we denote _{$L_{1}+l=\{a+l|a\in L_{1}\}$}.

(2) If$L_{1}\sim L_{2}$, then

we

have

$L_{1}=L_{2}\Leftrightarrow L_{1}\cap L_{2}\neq\emptyset\Leftrightarrow e_{m}(L_{1})=e_{m}(L_{2})$.

Lemma 2 Assume $L\subset \mathrm{C}$, $|L|=rm(m\geq 2)$. Then the followings

are

equivalent.

(1) Thereexists

a

monic polynomial$f(X)\in \mathrm{C}[X]$ ofdegree $m$with $|f(L)|=r$.

(2) There exists a decomposition of$L$, $L=L_{1}\cup\cdots\cup L_{r}$ with $|L_{k}|=m$, $L_{1}\sim$

. . . $\sim L_{r}$.

Using the above lemmas,

we can

prove the following Theorem.

Theorem 2 Let $\chi$ be

a

faithful character of

a

finite group $G$.

Set

$L=$

$\{\chi(g)|g\in G, g\neq 1\}$

.

Suppose that there exists a decomposition of $L$, $L=$

$L_{1}\cup\cdots \mathrm{U}L_{r}$ with _{$|L_{k}|=m\geq 2$}, _{$L_{1}\sim\cdots\sim L_{r}$}

.

Assume furtherthat each $L_{k}$

is algebraically closed. Then thereexists

a

monic $f(X)\in \mathrm{Z}[X]$ which satisfies

the following two conditions.

(i) $(G, \chi)$ is sharp of type $L$ $\doteqdot\Rightarrow$ $(G, f(\chi))$ is sharp oftype $f(L)$.

(ii) $f(L)=\{(-1)^{m-1}e_{m}(L_{1}), \cdots, (-1)^{m-1}e_{m}(L_{r})\}$

.

37

Example 3 Let (G,$\chi)$ be normalized sharp of type L $=$

{-1,0,1,2}.

Note

that L $=$

{-1,

$2\}\cup\{0,$

1}, {-1,

$2\}\sim$

{0,1}.

So L satisfies the conditions of

Theorem 2. If

we

put $f(X)=X^{2}-X$, then$(G, f(\chi))$is sharp oftype

{2,

0}

(but

not necessarily normalized). Using theclassification of sharp oftype $\{l, l+2\}$,

we

get $G=S_{5}$, $A_{6}$, $M_{11}$, Thus, $G$is

a

sharply 4-transitive group except $S_{4}$.

Example 4 $L=\{-1,0,2,3\}$ $=\{-1,3\}$ $\cup\{0, 2\}$ satisfies the conditions of

Theorem 2. Using $f(X)=X^{2}-2X$,

we can

reduce the determination of

L-sharp pairs to that of

_{3,,

0}-sharp

pairs. But unfortunately

we

do not have complete classification of $\{l, l+3\}$-sharp pairs.

Example 5 $L=\{-2, -1,0,2,3,4\}=\{-1,0,4\}$ $\cup\{-2,2,3\}$ satisfies the

conditions of Theorem 2. Using $f(X)=X^{3}-3X^{2}-4X$,

we can

reduce the

determination of $L$-sharp pairs to that of $\{0,$_$-12\}$-sharp pairs. But again we

do not have completeclassification of $\{l, l+12\}$-sharp pairs.

Remarks In Theorem 2, $f(\chi)$ is

a

generalized character of G and is not

necessarily character. $f(\chi)$ is not necessarily normalized,

even

if $\chi$ is so.

References

[A-N] D. Alvis and

S.

Nozawa, Sharpcharacterswithirrationalvalues, J. Math.

Soc. Japan, 48(1996))

567-591

[B] H. F.Blichfeldt, A theorem concerning the invariants of linearhomogeneous

groups with

some

applications to substitution groups, hans. Amer. Math.

Soc, 5(1984),

461-466

[C-K] P. J.Cameron andM. Kiyota, Sharp characters of finitegroups, J. Algebra

115(1988),

125-143

[C-K-K] P. J. Cameron, T. Kataoka and M. Kiyota, Sharp characters offinite

groups oftype

_{-1,

1},

J. Algebra 152(1992), 248-258

[N] S. Nozawa, Sharp characters of finite

groups

having prescribed values,